For accessibility, we recall all relevant concepts in Riemannian Geometry purely in terms of multivariate calculus and linear algebra and make no explicit mention of Riemannian metrics.
We refer the interested reader to other sources for the general theory (eg. [1]}.)
In this spirit, we take a smooth submanifold of \(\mathbb {R}^p\)  to be the preimage of 0 under a smooth function \(\mathbb {R}^p\rightarrow \mathbb {R}\)  which does not map any critical point to 0.
